---
tag: 1X1Y2AW4B0
kind: definition
name: Jacobson ring
msc: [13A15]
defines: 7EB64ZBXAT
uses: [0310REJJY8, 00MHHZ9SNE, 0PV74TR5FP, 437YRCNYRJ, XA0DCNH7WY, QQDN6EYQNJ]
sources:
  - url: https://en.wikipedia.org/wiki/Jacobson_ring
---

## Definition

Let $R$ be a [[commutative ring|0310REJJY8]].
The ring $R$ is a *Jacobson ring* if every [[prime ideal|00MHHZ9SNE]] $\mathfrak{p}$ is the intersection of the [[maximal ideals|0PV74TR5FP]] that contain $\mathfrak{p}$.

## Remarks

Example: $\Z$ is a Jacobson ring.
A nonzero prime ideal $p\Z$ is maximal, and the prime $0$ equals $\bigcap_p p\Z$, because a nonzero integer has only finitely many prime divisors.

Non-example: the localization $\Z_{(p)}$ is not a Jacobson ring.
Its only maximal ideal is $p\Z_{(p)}$, so the intersection of the maximal ideals containing the prime $0$ is $p\Z_{(p)} \ne 0$.

Equivalent forms: $R$ is Jacobson if and only if every [[radical ideal|437YRCNYRJ]] is an intersection of maximal ideals.
It is also equivalent that for every ideal $I$, the [[nilradical|QQDN6EYQNJ]] of $R/I$ equals its [[Jacobson radical|XA0DCNH7WY]].
Some texts use the name Hilbert ring.